A uniform metrical theorem in multiplicative Diophantine approximation

Letbe w sequences of differentiable functions defined in the rangeswhere aj, bj may be − ∞, + ∞ respectively. For each j and each m ≠ n letbe monotonic and let there be a constant K independent of m, n, j, θj such that

Estimates are found for the magnitude of overshoot, by a sequence of random variables, over an arbitrary boundary. If the sequence increments satisfy a so-called condition of asymptotic homogeneity and the boundary is asymptotically ``smooth, then the occurrence of the weak convergence to a limit shape (as the boundary is sent away) is established for the distribution of the overshoot value. As an application, we obtain a uniform (over the class of distributions) basic renewal theorem and determine the asymptotics of the average time of crossing a curvilinear border by the trajectories of asymptotically homogeneous Markov chains.

In May 1998, during the fifty-first voyage on board the research vessel Professor Vodyanitskii, a comparative study was conducted of the species diversity of green and purple sulfur bacteria in the water column of the chemocline zone at deep-sea stations and on the bottom surface of the Black Sea shallow regions. At three deep-sea stations, the accumulation of photosynthesizing bacteria in the chemocline zone at a depth of 85-115 m was revealed on the basis of the distribution of potential values of carbon dioxide light fixation. The location of the site of potential carbon dioxide light fixation suggests that the photosynthesis may be determined by the activity of the brown Chlorobium sp., revealed earlier at these depths. Enrichment cultures of brown sulfur bacteria were obtained from samples taken at the deep-sea stations. By morphology, these bacteria, assigned to Chlorobium sp., appear as nonmotile straight or slightly curved rods 0.3-0.5 x 0.7-1.2 microm in size; sometimes, they form short chains. Ultrathin sections show photosynthesizing antenna-like structures, chlorosomes, typical of Chlorobiaceae. The cultures depended on the presence of NaCl (20 g/l) for growth, which corresponds to the mineralization of Black Sea water. The bacteria could grow photoautotrophically, utilizing sulfide, but the Black Sea strains grew much more slowly than the known species of brown sulfur bacteria isolated from saline or freshwater meromictic lakes. The best growth of the strains studied in this work occurred in media containing ethanol (0.5 g) or sodium acetate (1 g/l) and low amounts of sulfide (0.4 mM), which is consistent with the conditions of syntrophic growth with sulfidogens. The data obtained allow us to conclude that the cultures of brown sulfur bacteria are especially adapted to developing at large depths under conditions of electron donor deficiency owing to syntrophic development with sulfate reducers. The species composition of the photosynthetic bacteria developing in the bottom sediments of shallow stations differed substantially from that observed at deep-sea stations. Pure cultures of the green Chlorobium sp. BS 1C and BS 2C (chlorobactin as the carotenoid), purple sulfur bacteria Chromatium sp. BS 1Ch (containing spirilloxanthine series pigments), and Thiocapsa marina BS 2Tc (containing the carotenoid okenone) were obtained from samples of sediments at shallow-water stations. Brown sulfur bacteria were absent in the sediment samples obtained from the Black Sea shallow-water stations 1 and 2.

Xylanases are hydrolytic enzymes with wide range of applications in food processing, bleaching of pulp in paper manufacturing industry, bio-conversion of biomass wastes to fermentable sugars and enhancing nutrient digestibility in animal feeds. The optimization of growth conditions and evaluation of an appropriate substrate as carbon source among cassava peels, corn cobs, wheat bran and rice husk on xylanase production by novel strain of Aspergillus tubingensis under Solid State Fermentation (SSF) was investigated. The fungal isolate was identified based on ribosomal RNA gene and ITS gene sequencing analysis as Aspergillus tubingensis. Results showed that Corn cobs had the highest xylanase production among the four substrates. Corn cobs recorded the highest value of xylanase production at pH of 6.0 (107.97 U/g), after incubation period of 72 hour (111.23 U/g), at temperature of 30oC (44.26 U/g) and at ratio 1:3 (45.68 U/g). The optimum growth conditions for xylanase production by Aspergillus tubingensis were: pH 6.0, incubation period of 72 hours, temperature of 30oC and substrate concentration of 1:3 (4g of substrate per 12ml of fermentation medium). Corn cobs showed the highest yield of xylanase activity (111.23±0.31 U/g), followed by Rice husk (101.91±0.72 U/g), Wheat bran (89.30±1.16 U/g) and Cassava peel (87.03±0.57 U/g). In conclusion, among the various agro residues that were used as carbon sources, Corn cobs had maximum xylanase activity. Various culture conditions were optimized by using one factor at a time method and the maximum xylanase production was obtained at pH of 6.0, incubation period of 72 hour, temperature of 30oC and substrate concentration of 1:3 under solid state fermentation. It is therefore suggested that some other notable environmental and fermentation factors that influence metabolism-mediated production yields of xylanase like aeration, agitation, carbon and nitrogen sources, metal ion requirement, inoculum size etc. should be optimized for maximum production of enzyme. Keywords: Solid State Fermentation, Cassava peels, Corn cobs, Wheat bran, Rice husk

Simultaneous asymptotic diophantine approximations to a basic of a real cubic number field

The present paper is defining and studding a modification of the general family sequence of summation Baskakov-type operators. This modification is preserved that the functions and where is fixed. We show that the uniform convergence theorem of this sequence by using the modulus of continuity to the function being approximated. Finally, we introduce the asymptotic formula for the Voronovskaya-type theorem

IntroductionCoastal nations heavily rely on their sea areas to achieve sustainable socioeconomic development. Sea areas offer abundant marine ecosystem products and services that are difficult to substitute with industrial goods. Although considerable research has been performed to evaluate the contribution offrom the marine ecosystems to human well-being, systematic and measurable indices are lacking.MethodsTo address this knowledge gap, ecosystem service theory was used to establish a framework for assessing the gross ecosystem product (GEP) of a sea area, which represents the total monetary value of final ecosystem products directly and indirectly used by people within a specific administrative jurisdiction in one year. To evaluate marine GEP, three primary indices, i.e., material products, regulating products, and cultural products, were employed, and they consist of eight secondary indices. Material products in the sea area include aquatic products and oxygen products; regulating products include carbon sequestration, climate regulation, waste treatment, and coastal protection; and cultural products include leisure and recreation, and seascape-added value. The marine GEP was calculated for the administrative sea areas of Changdao County, which is located in the Yellow Sea-Bohai Sea transitional zone.Results and discussionOur analysis indicated that from 2017 to 2019, the marine GEP in Changdao ranged between 32.59 and 33.35 billion Chinese yuan (CNY). Regulating products contributed two-thirds of this total value. However, over the three-year period, the marine GEP underwent a 5.64% decrease in Changdao. Specifically, the value of material products increased by 38.85% while that of regulating and cultural products decreased by 9.16% and 15.87%, respectively. Notably, the spatial distribution of ecosystem product values in Changdao displayed two prominent trends: (1) higher values along the coastal sea areas of islands and lower values in offshore sea areas; and (2) higher values in coastal areas of southern islands and lower values in coastal areas of northern islands. Marine GEP may serve as a comprehensive indicator for assessing the sustainability of marine ecosystem. Combining the assessment of both marine and terrestrial GEP facilitates an understanding of how marine and terrestrial ecosystems interact in coastal regions. Furthermore, the combined use of marine GEP and GDP helps better evaluate and sort the level of green development of coastal nations and regions.

By introducing a ubiquity property for rectangles, we prove the mass transference principle from rectangles to rectangles, i.e., if a sequence of rectangles forms a ubiquity system (a full measure property), then the limsup set defined by shrinking these rectangles to smaller rectangles has full Hausdorff measure or to say transfer a full measure property to a full Hausdorff measure property for brevity. The limsup sets generated by balls or generated by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet's theorem, the other follows from Minkowski's theorem. So the result sets up a general principle for the Hausdorff measure theory for high dimensional Diophantine approximation which, together with the landmark work of Beresnevich & Velani in 2006 where a transference principle from balls to balls is established, gives a coherent Hausdorff measure theory for metric Diophantine approximation. The dimensional theory for limsup sets generated by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods or even their generalizations fail to work.

In this paper, we study the value distribution of the Sudler product ∏ n=1 N |2sin(πnα)| and the Diophantine product ∏ n=1 N (2e∥nα∥) for various irrational α, as N ranges in a long interval of integers. At badly approximable irrationals these products exhibit strong concentration around N 1/2 , and at certain quadratic irrationals they even satisfy a central limit theorem. In contrast, at almost every α we observe an interesting anticoncentration phenomenon when the typical and the extreme values are of the same order of magnitude. Our methods are equally suited for the value distribution of Birkhoff sums ∑ n=1 N f(nα) for circle rotations. Using Diophantine approximation and Fourier analysis, we find the first and second moment for an arbitrary periodic f of bounded variation, and (almost) prove a conjecture of Bromberg and Ulcigrai on the appropriate scaling factor in a so-called temporal limit theorem. Birkhoff sums also satisfy a central limit theorem at certain quadratic irrationals.

In the present paper we give an improvement of a previous result of the paper [M. M. Skriganov. Ergodic theory on $SL(n)$, diophantine approximations and anomalies in the lattice point problem. Inv. Math.132(1), (1998), 1–72, Theorem 2.2] on logarithmically small errors in the lattice point problem for polyhedra. This improvement is based on an analysis of hidden symmetries of the problem generated by the Weyl group for $SL(n,\mathbb{B})$. Let $UP$ denote a rotation of a given compact polyhedron $P\subset\mathbb{B}^n$ by an orthogonal matrix $U\in SO(n)$, $tUP$ a dilation of $UP$ by a parameter $t>0$ and $N(tUP)$ the number of integer points $\gamma\in\mathbb{Z}^n$ which fall into the polyhedron $tUP$. We show that for almost all rotations $U$ (in the sense of the Haar measure on $SO(n)$) the following asymptotic formula \[ N(t\UP)=t^n{\rm vol} P+ O((\log t)^{n-1+\varepsilon}),\quad t\to\infty, \] holds with arbitrarily small $\varepsilon>0$.

Agricultural statistical data enable the detection and interpretation of the development of agriculture and the food supply situation over time, which is essential for food security evaluation in any country. Based on the historical agricultural statistics, this study produces a long spatial time-series with annual production values of three cereals relevant to global food security—wheat, maize, and rice, aiming to provide geographical and historical perspectives. Therefore, we reconstructed past and current production patterns and trends at the district level over 169 years, which supported a space–time cross-reading of the general characteristics of the regional agricultural production value distributions and relative densities in Portugal. Particularly, the production trends of wheat, maize, and rice showed three different situations: growth (maize), stability (rice), and decline (wheat). For decades, maize and wheat production alternated, depending on agricultural years and political aspects, such as the Wheat Campaign (1929–1938). The changes over time presented a pattern that, in the case of these three cereals, enabled a clear division of the country into major regions according to cereal production. Overall, maize and rice, both grown on irrigated croplands, presented a similar pattern in some regions of Portugal, mainly the central region. In this study, a preliminary analysis was presented and related to successive public policies; however, notably, there are more lessons to be learned from this long spatial time-series.

Background: Multi-detector computed tomography (MDCT) is the preferred modality for follow-up of paediatric neurosurgery patients. Serial imaging, however, has the disadvantage of an ionising radiation burden, which may be mitigated using the ‘as low as reasonably achievable’ (ALARA) principle. Objectives: The primary objectives were to determine the radiation dose exposure in paediatric patients subjected to MDCT imaging following neurosurgery and to compare these values with references in current literature. Our secondary objective was to assess the relationship between radiation dose and clinical scenario.Method: Retrospective descriptive data were collected from all paediatric postsurgical patients (n = 169) between the ages of 0 and 12 years who had their first followed-up scan in the year 2010 and were followed up for six months or less. Dose-length product (DLP) and current-time product were collected from the picture archiving and communication system. Demographic data including radiology reports were collected from the hospital information system. The effective doses (ED) were calculated from the corresponding DLP using age-adjusted conversion factors. For purposes of comparison with other studies, median dosimetric values were calculated and the children were grouped into three age ranges, namely younger than 3 years, 3–7 years and 8–12 years old.Results: The highest median radiation doses were noted in patients being followed-up for intracranial abscesses (1183 mGy cm) in the 8–12 year age group, most of whom were female. The lowest radiation doses were for intracranial shunt follow-ups (447 mGy cm). Median values for DLP, ED and current-time product (mAs) were comparable to reference doses in all three age groups. However, our study showed a much broader distribution of values with higher upper limits relative to reference values. Indications for follow-up included shunts (n = 110; 65%), intracranial abscess (n = 31; 18%), subdural haematoma (n = 13; 8%) and tumour (n = 6; 4%). Head trauma only accounted for 5% of the cases.Conclusion: The median radiation doses measured were comparable to values in literature and therefore deemed acceptable. The wider dose distributions of all three dosimetric parameters (DLP, ED and mAs) were attributed to inappropriate use of scan length and reference effective mAs. Adherence to recommended scan length protocols should be encouraged. Evaluation of the current use of reference effective mAs is needed and will require a separate study to determine the smallest value that can be used without compromising image quality. Further dose reductions could be achieved by omission of unenhanced scans in the follow-up of intracranial abscesses. It is recommended that diagnostic reference levels specific to South African clinical scenarios be developed to make local dosimetric audits more relevant.

Диофантовы экспоненты являются одними из самых простых количественных характеристик, отвечающих за аппроксимационные свойства линейных подпространств евклидова пространства. Данный обзор посвящeн описанию современного состояния раздела теории диофантовых приближений, изучающего диофантовы экспоненты и соотношения, которым они удовлетворяют. Мы обсуждаем классические диофантовы экспоненты, возникающие в задаче приближения нуля набором значений нескольких линейных форм в целых точках, их аналоги в теории диофантовых приближений с весами, мультипликативные диофантовы экспоненты, а также диофантовы экспоненты решeток. Особое внимание уделяется принципу переноса. Библиография: 99 названий.

<p>The main purpose of this paper is using the elementary method to study the value distribution property of the Smarandache multiplicative function, and give an interesting asymptotic formula for it.</p>

Rational points near planar curves and Diophantine approximation